For Form 8938, the instructions indicate that you should use their quarterly exchange for US TreasuryBureau of the Fiscal Service exchange rate which as you indicated is published quarterly.
In most cases, you must use the U.S. Treasury Bureau of the Fiscal Service foreign currency exchange rate for purchasing U.S. dollars. You can find this rate on www.fiscal.treasury.gov/fsreports/rpt/treasRptRateExch/treasRptRateExch_home.htm. If no U.S. Treasury Bureau of the Fiscal Service exchange rate is available, you must use another publicly available foreign currency exchange rate for purchasing U.S. dollars and disclose the rate on Form 8938.
Currency determination date.
Use the currency exchange rate on the last day of the tax year to figure the maximum value of a specified foreign financial asset or the value of a specified foreign financial asset for the purpose of determining the total value of your specified foreign financial assets to see whether you have met the reporting threshold. Use this rate even if you sold or otherwise disposed of the specified foreign financial asset before the last day of the tax year.
